Now I’m not sure what the ’it’ of

indeed it’s not

refers to. Or the ’this’ of

I already said this.

I was expecting $\mathbf{H}^I$ **not** to be an infinitesimal thickening of $\mathbf{H}$. I was wondering if this fact could already be seen from the failure of $\infty Grpd^I$ to be infinitesimally cohesive over $\infty Grpd$.

It’s part of my transporting thickenings idea. Were $\mathbf{H}^I$ a thickening over $\mathbf{H}$ we could transport it to $\infty Grpd^I$ over $\infty Grpd$. But we know we can’t.

But maybe the transporting idea isn’t sound.

]]>Yes, indeed it’s not. Sorry, I thought I already said this. In a way it is the finite length of the interval $(0 \to 1)$ that prevents it, for it forces the neighbouring adjoints to be different, which for $T \mathbf{H}$ coincide.

By the way, I have now added on p. 256 more remarks previewing the way that the homotopy cofiber of $\infty$-toposes along a differential cohesion inclusion produces the underlying infinitesimal cohesion, here https://dl.dropboxusercontent.com/u/12630719/cohesivedocument131024.pdf.

This is quite probably true in full generality, but right now I just show it in the examples.

]]>I guess I was hoping that exponentiation by $I$ is simple enough that the failure of $\infty Grpd^I$ to be infinitesimally cohesive would imply that $H^I$ is not differentially cohesive for cohesive $H$.

]]>I need to think about those higher jet toposes more in order to answer, but meanwhile just one comment to clear up terminoloy (since I may have cause a mixup here)

I am saying

“infinitesimally cohesive” if $\Pi \simeq \flat$

“differentially cohesive” if the cohesion is factored through a geometric embedding.

Now if $\mathbf{H}$ is infinitesimally cohesive over $\infty\mathrm{Grpd}$, then the inclusion $\infty Grpd \hookrightarrow \mathbf{H}$ also exhibits $\mathbf{H}$ as being differentially cohesive over $\infty Grpd$. But the converse does not in general hold.

]]>Yes I knew that it wasn’t differentially cohesive as the modalities don’t equate. But the original question was then where in the tower of cohesive jet toposes does differential cohesiveness fail. If $J^1 = T$ is differential, does it already fail at $J^2$?

]]>Oh, I see. Yes, my fault. Fixed now.

Concerning differentially cohesive: ah, now I see what you mean. No, not over $\mathbf{H}$, unless I am missing something. True.

What makes one kind of cohesive homtopy types $X$ be “differential cohesive” over another $x$ is if they are “infinitesimal thickenings” of that latter, meaning that you cannot see the thickening of $X$ when homming into it out of $x$; something non-trivial only happens the other way around, homming $X$ into $X$ sees “tangents” in $x$.

]]>Google must be picking up similar misspellings then. It’s double c and double m: accommodate. (Latin: ad-commodus, probably in turn con-modus)

Yes $\mathbf{H}^I$ is cohesive, but is it **differentially** cohesive?

That would be great if you would look over especially the physics slides (when they’re ready).

]]>Thanks!

Concerning “accomodate”, isng’t that correct? I checked with Google and Google seems to agree? Did you maybe make a reverse typo when reporting a typo?

Concerning $\mathbf{H}^I$: this is in fact cohesive over $\mathbf{H}$, but my section on it doesn’t mention that (yet). Back when I wrote that section I used $\mathbf{H}^I$ for other purposes than giving twisted cohomology an internal home. It is now only through the company of the flashy $T \mathbf{H}$ that its superficially boring cousin $\mathbf{H}^I$ is getting some recognition, too.

Concerning your talk: interesting! If you have any preliminary notes, I’d be happy to look through them and return a tiny bit of the favor.

BTW, that same week I will be speaking in Edinburgh on “Higher toposes of laws of motion”.

]]>That made me see

’accomodate’ twice

’projetion’

Since $T \mathbf{H}$ is differentially cohesive over a cohesive $\mathbf{H}$, but $\mathbf{H}^{\Delta[1]}$ is not, at what point in the $J^n(\mathbf{H})$ interpolating between them does this property go?

By the way, it’s not just altruism that has me reading your book. I’ve promised to give a talk with the grand title Homotopy Type Theory: a revolutionary language for philosophy of logic, mathematics, and physics?. I’m nervous about getting the physics part up to scratch. Think I might talk about covariance. Maybe also something on quantization.

]]>I have now added some notes on this to section 4.1.2 of https://dl.dropboxusercontent.com/u/12630719/cohesivedocument131024.pdf

]]>Have now finally contacted Georg Biedermann. He kindly points out his articles where he constructs model categories of n-excisive functors, which are homotopy toposes in Charles’s terminology, hence present the $\infty$-toposes that we are discussing.

While the Lab is unresponsive, I’ll record these reference here:

Georg Biedermann, Boris Chorny, Oliver Röndigs,

*Calculus of functors and model categories*, Advances in Mathematics 214 (2007) 92-115 (arXiv:math/0601221)Georg Biedermann, Oliver Röndigs,

*Calculus of functors and model categories II*(arXiv:1305.2834v2)

About those two flat tangent connections mentioned in #29, Goodwillie says in the abstract for a 2005 workshop

their difference is the tensor ﬁeld known as smash product of spectra.

He continues

]]>I will say something about higher-order jets and about differential operators. I cannot make much sense of differential forms (except 0-forms and 1-forms), but I may talk about them anyway.

I see, thanks for the information. I suppose it is about time that I contact Georg Biedermann.

]]>Urs 51. I’m probably hallucinating. I don’t know where he would have said this, though it is true.

Goodwillie does briefly discuss parameterized spectra (in Calculus 1, remark 1.6). Here he is thinking about functors $\mathrm{Top}/X\to \mathrm{Top}_*$ (it would be the same I think if you consider functors $X\backslash \mathrm{Top}/X\to \mathrm{Top}$), and in this case reduced $1$-excisive functors are the same as parameterized spectra over $X$.

However, this is a completely different appearence of parameterized spectra, than the one I discussed above. (Basically, an object of $\mathcal{F}_1$ is a parameterized spectrum over the space $T=F(*)$ which is the *value* of $F$ at the terminal object of the domain; 1-excisive functors on the slice category $X\backslash\mathrm{Top}/X$ want to be parameterized over $X$, which is itself the terminal object of the domain.)

Urs 51: I’m not sure. I think it was in Calculus 3, but that would have been a preprint version I saw many years ago.

Urs 50: I told Jacob that, but I’d learned it from Biedermann.

]]>Goodwillie showed that […] the 1-excisive functors F equipped with an identification of P0F with the space X, is the same as parameterized spectra over X.

Maybe you can save me a minute: which page is this statement on?

]]>Oh, and in remark 7.1.1. 11 right below Lurie attributes the statement of your #47 above to you. :-)

Is this in print anywhere? Do you say this in “Homotopy toposes”? (I don’t remember having seen it there, but I may have forgotten.)

]]>Fully explicitly, it is in theorem 7.1.1.10 of Lurie’s “Higher Algebra”.

I suppose I should read that book someday.

]]>Charles,

thanks for pushing this further!

This would be theorem 1.8 in “Calculus III”, I suppose.

Fully explicitly, it is in theorem 7.1.1.10 of Lurie’s “Higher Algebra”.

]]>I understand things like this:

I write $\mathrm{Top}$ for the homotopy theory of spaces (i.e., $\infty$-groupoids). Let $\mathcal{F}=\mathrm{Func}^{\mathrm{filt}}(\mathrm{Top}_*,\mathrm{Top})$ denote the homotopy theory of functors from pointed spaces to spaces which commute with all filtered colimits. We can identify a full subtheory $\mathcal{F}_n\subset \mathcal{F}$ of $n$-excisive functors.

Goodwillie (in “Calculus 3”) has given an explicit formula to construct the left adjoint $P_n\colon \mathcal{F}\to \mathcal{F}_n$ to the inclusion $\mathcal{F}_n\subset \mathcal{F}$, called “$n$-excisive approximation”. It’s immediate from Goodwillie’s formula that $P_n$ commutes with finite homotopy limits, and therefore each $\mathcal{F}_n$ is an $\infty$-topos. Thus we obtain a tower of $\infty$-topoi $\mathcal{F}\to \cdots \to \mathcal{F}_n\to\mathcal{F}_{n-1}\to \cdots\to \mathcal{F}_1\to \mathcal{F}_0$.

$\mathcal{F}_0$ consists of constant functors, and so is equivalent to $\mathrm{Top}$, while $\mathcal{F}_1$ is $1$-excisive functors; Goodwillie showed that $(\mathcal{F}_1)^{P_0=X}$, the $1$-excisive functors $F$ equipped with an identification of $P_0F$ with the space $X$, is the same as parameterized spectra over $X$.

“Intrinsic” characterizations of the $\mathcal{F}_n$ for $n\gt1$ seem to be much harder to get at. Probably one can say that $(\mathcal{F}_n)^{P_0=X}$ is equivalent to “parameterized reduced $n$-excisive functors over $X$”, where “reduced $n$-excisive functors” are $(\mathcal{F}_n)^{P_0=*}$. It is hard to give a characterization of $(\mathcal{F}_n)^{P_0=*}$ for $n\gt1$ that is satisfying, though Arone and Ching now have a decent answer to this, I think, in terms of comonads on spectra.

**Correction.** When I originally wrote this, I wrote $\mathrm{Func}^{\mathrm{filt}}(\mathrm{Top},\mathrm{Top})$ for the definition of $\mathcal{F}$, a category of functors on unbased spaces, thus perpetuating a standard confusion in this subject. There is perfectly good theory of $n$-excisive approximations for functors on unbased spaces, and it gives rise to a nice tower of $\infty$-toposes (that construction is very general)— but this tower is somewhat *different* than the one I actually described above. For instance, “unbased $1$-excisive functors” are the same thing as “parameterized spectra $E\to X$ equipped with a section”.

Urs #43, I don’t quite see how things go here. In the $n = 1$ case, the idea is that there’s a connection between 1-excisive functors and the tangent $(\infty, 1)$-category construction.

For the latter, I see how that diagram seq gets used to define parameterized spectra, and I see how that delivers the fiberwise stabilization of the codomain fibration, $Stab(Func(\Delta[1],C) \to C)$.

So how does this relate to 1-exciveness, which concerns taking pushout squares to pullback squares? Is the relation via the squares of seq?

If so, when you say we do the same for $k$-excisive functors, is it that seq gets replaced by a higher dimensional with $k+1$-cubes instead of squares? And what is the equivalent of $Stab(Func(\Delta[1],C) \to C)$? As I had it in #40?

]]>With all this dash to the $(\infty, 1)$ case, I’d forgotten Urs had already being doing tangents for $(n, n)$-categories way back here and here. Did the extra generality get used, or did it mostly boil down to the $(n, 1)$ case?

]]>I was thinking there should be a multivariate calculus of functors, and of course there is, see sec. 6.2. So we have a notion of **n**-excisive for **n** = $(n_1, ..., n_m)$.

But presumably there should be a multivariate jet construction. Thinking in SDG terms, we need Kock’s $D_k(n) = [(x_1, ..., x_n) \in R^n|$ product of any $k+1$ is zero].

]]>Mike, the idea is to phrase Joyal’s observation in terms of 1-excisive functors and then observe that it goes through for $k$-excisive functors, too.

]]>